Theorem sbcth2

Description: A substitution into a theorem. (Contributed by NM, 1-Mar-2008) (Proof shortened by Mario Carneiro, 13-Oct-2016)

	Ref	Expression
Hypothesis	sbcth2.1	$$⊢x\in B\to \phi$$
Assertion	sbcth2	$$⊢A\in B\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi$$

Proof

Step	Hyp	Ref	Expression
1		sbcth2.1	$$⊢x\in B\to \phi$$
2	1	rgen	$$⊢\forall x\in B\phi$$
3		rspsbc	$$⊢A\in B\to \left(\forall x\in B\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)$$
4	2 3	mpi	$$⊢A\in B\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi$$